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Aurifeuillean Factorization

A factorization of the form

2^(4n+2)+1=(2^(2n+1)-2^(n+1)+1)(2^(2n+1)+2^(n+1)+1).
(1)

The factorization for n=14 was discovered by Aurifeuille, and the general form was subsequently discovered by Lucas. The large factors are sometimes written as L and M as follows:

2^(4k-2)+1=(2^(2k-1)-2^k+1)(2^(2k-1)+2^k+1)
(2)
3^(6k-3)+1=(3^(2k-1)+1)(3^(2k-1)-3^k+1)(3^(2k-1)+3^k+1),
(3)

which can be written

2^(2h)+1=L_(2h)M_(2h)
(4)
3^(3h)+1=(3^h+1)L_(3h)M_(3h)
(5)
5^(5h)-1=(5^h-1)L_(5h)M_(5h),
(6)

where h=2k-1 and

L_(2h),M_(2h)=2^h+1∓2^k
(7)
L_(3h),M_(3h)=3^h+1∓3^k
(8)
L_(5h),M_(5h)=5^(2h)+3·5^h+1∓5^k(5^h+1).
(9)

See also

Gauss's Cyclotomic Formula

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References

Brillhart, J.; Lehmer, D. H.; Selfridge, J.; Wagstaff, S. S. Jr.; and Tuckerman, B. Factorizations of b-n+/-1, b=2, 3, 5, 6, 7, 10, 11, 12 Up to High Powers, rev. ed. Providence, RI: Amer. Math. Soc., pp. lxviii-lxxii, 1988.Riesel, H. "Aurifeullian Factorization" in Appendix 6. Prime Numbers and Computer Methods for Factorization, 2nd ed. Boston, MA: Birkhäuser, pp. 309-315, 1994.Wagstaff, S. S. Jr. "Aurifeullian Factorizations and the Period of the Bell Numbers Modulo a Prime." Math. Comput. 65, 383-391, 1996.

Referenced on Wolfram|Alpha

Aurifeuillean Factorization

Cite this as:

Weisstein, Eric W. "Aurifeuillean Factorization." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/AurifeuilleanFactorization.html

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